Small advice on doing Maxwell's covariant form

Question

Hello fellow physicists!

I really enjoyed that Carroll (Spacetime and Geometry) included how tensors can be used to rewrite Maxwell's equations.

→Firstly rewriting the usual in tensor/index notation:

→Then showing some steps (which were simple enough to deduce what was going on and what I had to do to arrive to the result) to unify the first 2 equations:

→He then presents the idea that we can do the same for the last 2 equations, and we should arrive to the following:

→I would really like to do the derivation, yet I'm a bit lost. I tried different approaches yet I arrived to somehow nonsense. It is also unclear to me how to move from latin to greek letters in this problem without messing with the Levi-Civita symbol.

→ I just need a small push to do the rest by myself.

Thanks for any advice you can give. I tried looking for similar questions, yet I guess each has its own notation and such.

Answer 1

Equation (1.98) reduces to $0=0$ if any two of the free indices are the same. So consider how they can all be different: either they are three different spatial indices, or two are different spatial indices and the third is temporal. The former gives the fourth 3D Maxwell equation, and the latter gives the third.

In more detail, when the three indices are three different spatial indices, they are obviously 1, 2, and 3. Equation (1.98) becomes

$$\partial_1F_{23}+\partial_2F_{31}+\partial_3F_{12}=0$$

or

$$\partial_1B^1+\partial_2B^2+\partial_3B^3=0.$$

This is

$$\partial_iB^i=0.$$

When one of the indices is temporal, the two spatial indices can be 1 and 2,or 2 and 3, or 3 and 1. Let's do the first case. We get

$$\partial_0F_{12}+\partial_1F_{20}+\partial_2F_{01}=0$$

or

$$\partial_0B^3+\partial_1E_2-\partial_2E_1=0.$$

This is the $i=3$ component of the equation

$$\epsilon^{ijk}\partial_jE_k+\partial_0B^i=0.$$

The other two ways to choose the spatial indices give the other two components of the equation.

Answer 2

Probably not the answer you were hoping for, because you might want to operate with the field tensor directly.

But since I also find the transition from 3D Levi-Civita to 4D a little cumbersome and error-prone, I would just start from the fact that Maxwell's equations already fully imply that you can write the field tensor in terms of a vector potential $$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$$ Then the homogeneous equations become rather trivial in 4D: $$\epsilon_{\kappa\lambda\mu\nu}\partial^\lambda F^{\mu\nu}=\epsilon_{\kappa\lambda\mu\nu}\partial^\lambda (\partial^\mu A^\nu-\partial^\nu A^\mu)=2\epsilon_{\kappa\lambda\mu\nu}\partial^\lambda \partial^\mu A^\nu=0$$ because it is symmetric and antisymmetric at the same time in the indices $\lambda,\mu$.

I think it should be easy to rewrite this without Levi-Civita at all.